Notes on exceptional signed graphs - Stanić, Zoran (Ars mathematica contemporanea, 2020)
A connected signed graph is called exceptional if it has a representation in the root system ?$E_8$?, but has not in any ?$D_k$?. In this study we obtain some properties of these signed graphs, mostly expressed in terms of those that are maximal with a fixed number of eigenvalues distinct from ?$-2$?. As an application, we characterize exceptional signed graphs with exactly 2 eigenvalues. In some particular cases, we prove the (non-)existence of such signed graphs.
Keywords: adjacency matrix; exceptional signed graph; izjemni označeni graf; korenski sistem; least eigenvalue; matrika sosednosti; najmanjša lastna vrednost; označeni linijski graf; razgradnja označenega grafa; root system; signed graph decomposition; signed line graph